A Mixed Volume Element Modified With Characteristic Fractional Step Difference Method for the Compressible Multicomponent Displacement and Its Numerical Analysis

A three-dimensional compressible problem with different components is fundamental in numerical simulation of enhanced oil recovery. The mathematical model consists of a parabolic equation for the pressure and a convection-diffusion system for the concentrations. The pressure determines Darcy velocity and plays an important role during the whole physical process. A conservative mixed volume element is used to discretize the flow equation, and improves the computational accuracy of Darcy. The concentrations are computed by the modified characteristic fractional step difference scheme, thus numerical dispersion and nonphysical oscillations are eliminated. The whole three-dimensional computation is accomplished effectively by solving three successive one-dimensional problems in parallel, where the speedup method is used and the work is decreased greatly. Based on the theory and special techniques of a priori estimates of partial differential equations, an optimal second error estimates in L2-norm is concluded. This work concentrates on the model, numerical method and convergence analysis for modern oil recovery.


Introduction
An important tool is used usually in modern oil recovery that high-pressure pump drives water into oil reservoir and displaces the crude oil from production wells. The 3-D problem is studied carefully in this paper. In modern oil-gas development process, a new enhanced oil recovery (chemical oil recovery) technique is adopted to displace the remnant crude oil from oil reservoir. The compressibility of fluid and the multicomponent must be considered in numerical simulation of enhanced oil recovery, otherwise numerical simulation probably distorts the truth. Douglas and other researchers first put forward the mathematical model for a compressible miscible multicomponent flow, and discuss the method of characteristic finite element and the method of characteristic mixed finite element.
Here p(X, t) is the pressure, and c α (X, t) is the αth concentration for α = 1, 2, · · · , n c . n c is the number of components.

Because of n c
The compressibility and the multicomponent are considered in simulating the physical displacement of enhanced oil recovery. The computational work is large-scaled on a three-dimensional region and a long time interval. Millions of variable nodes are involved, so some standard numerical methods are invalid. Then, an effective fractional step method is proposed (Ewing, 1983;Yuan, 2013;Shen, Liu & Tang, 2002). Peaceman and Douglas discuss the fractional step difference to solve a two-dimensional problem but argue the stability and convergence only for a constant coefficient case by using Fourier analysis (Peaceman, 1980;Douglas & Gunn, 1963,1964. Yauenko, Samarskii and Marchuk study this method (Yanenko, 1967;Marchuk,1990), and Yuan gives a characteristic fractional step difference for a two-dimensional problem and present convergence analysis (Yuan,1999(Yuan, ,2003. From the previous work, the authors propose a mixed volume element modified with the characteristic fractional step differnece for a three-dimensional compressible multicomponent displacement problem. The flow equation is treated by a conservative mixed volume element, and the accuracy of Darcy velocity is improved one order. The concentration vector is computed by a second characteristic fractional step difference scheme. Since the method of characteristics is used, numerical dispersion, oscillations and the computational complexity are overcome. A large time step could be adopted without any loss of accuracy. The whole computation is accomplished by solving three successive one-dimensional problems in parallel. A speedup method is used, and the computation is decreased greatly. By the variation, energy norm analysis, the usage of different meshes, piecewise product threefold quadratic interpolation (27-point interpolation) (Ciarlet,1978), decomposition of high-order difference operators and the interchangeability of different operators, we obtain an optimal error estimates in L 2 -norm. This work maybe gives some valuable references in the research on numerical simulation of modern oil recovery such as model analysis, numerical method, mechanism study and engineering software (Douglas, 1983b;Ewing,1983;Yuan,2013;Shen, Liu & Tang, 2002).
The problem of (1)-(4) is regular, In this paper, the symbols M and ε denote a generic positive constant and a generic small positive number, respectively. They may have different definitions at different places.

Notation and Preliminaries
Two different partitions are introduced to formulate the scheme. For simplicity, take Ω = [0, 1] 3 . Define the first partition for the flow equation, , h y and h z can be defined similarly for j = 1, 2, · · · , N y and k = 1, 2, · · · , N z . Let Ω i jk = I x i × I y j × I z k and h p = (h 2 x + h 2 y + h 2 z ) 1/2 . Suppose that the partition is regular (see Fig. 1 as an example illustration).
Proof. It requires to prove that ||ŵ Similarly, other inequalities are proved.

Lemma 3 For q ∈ S h , For q ∈ S h , there exists a number M independent of q, h such that
and use the Cauchy inequality, we have Multiply the both sides by h x,i+1/2 h y j h z k , and make the sum, The proof ends.

The Procedures
The flow equation (1) is rewritten in a normal form to construct the mixed volume element, Let P, U and C denote the numerical solutions of p, u and c, respectively. Here C is computed by a threefold quadratic interpolation on the refined partition Ω h (Yuan, 2013;Ciarlet, 1978). Recalling the notation and preliminary properties, we obtain the mixed volume element procedures for the pressure and Darcy velocity (Russell, 1995;Weiser & Wheeler, 1988;Jones, 1995), The flow shown in Eq.
(2) moves along the characteristics, so the method of characteristics is used to approximate the hyperbolic term. This treatment has strong stability and high accuracy. A large time step may be used during the computations. Let ψ(X, u) = [ϕ 2 (X) + |u| 2 ] 1/2 and ∂ ∂τ = ψ −1 {ϕ ∂ ∂t + u · ∇}. A backward difference quotient is given for the derivative along the characteristic direction, Then, the characteristic fractional step difference scheme is concluded for Eq. (2), where C n α (X)(α = 1, 2, · · · , n c − 1) is determined by a threefold quadratic interpolation of {C n α,i jk } at twenty-seven points Initial approximations, The composite procedures run as follows. From (17) and the elliptic projections {Ũ 0 ,P 0 } (seen in the next section), take U 0 =Ũ 0 and P 0 =P 0 . Using the scheme (13) and the method of conjugate gradient, we get {U 1 , P 1 }. Then, from (14)- (16) and the speedup algorithm, {C 1 α }, α = 1, 2, · · · , n c −1} is computed in parallel. Repeat the computations as above, we could all the numerical solutions. From (C), we find that the solutions exist and are unique.

Convergence Analysis
Introduce the elliptic projections first.
Suppose that the problem of (1) and (2) is positive definite (C) and properly regular (R). From the theory of Weiser and Wheeler (Weiser & Wheeler, 1988;Jones, 1995), it is easy to see that the solutions of (18),Ũ andP exist and are estimated in Lemma 5.

Conclusions and Discussions
In this paper, the authors present a mixed volume element modified with characteristic fractional step difference, and discuss its numerical analysis. The mathematical model, physical interpretation and academic research are introduced in §1. Two different partitions (coarse and refined) are given for defining the scheme, and some preliminary properties are stated for theoretical analysis in §2. In §3, the authors formulate the procedures based on the combination of the mixed volume element and modified characteristic fractional step difference. The mixed volume element has the conservative nature for the flow. The characteristic fractional step difference is illustrated to compute the concentration vector without numerical oscillations. A large time step is adopted and the speedup algorithm is used during the whole computation. The three-dimensional work is accomplished effectively by solving three successive one-dimensional problems. In §4, using the variation, energy norm analysis, the usage of different meshes, piecewise product threefold quadratic interpolation, decomposition of high-order difference operators and the interchangeability of different operators, we obtain an optimal error estimates in L 2 -norm. There are several interesting conclusions.
• This discussion considers the compressibility and the multicomponent, thus the numerical simulation is possibly consistent with the truth.
• The composite procedures could be carried out in three-dimensional complicated region.
• The mixed volume element has the nature of conservation, and improves the computational accuracy of Darcy one order. This property is important in numerical simulation of two-phase seepage displacement.
• The concentration vector is computed by the characteristic fractional step difference, where a large time step is used and second accuracy is preserved. Thus, this parallel algorithm could be carried out to complete numerical simulation on parallel computers.